Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T05:16:00.179Z Has data issue: false hasContentIssue false
  • English
  • Français

An Iterative Technique for Minimising a Double Integral with Applications in Elasticity

Published online by Cambridge University Press:  04 July 2016

J. P. H. Webber
J. P. H. Webber*
Affiliation:
Department of Civil Engineering, Royal Military College of Science, Shrivenham
Get access Rights & Permissions [Opens in a new window]

Extract

The solution of a problem in plane elasticity can be associated with a minimum energy principle, involving the minimisation of a double integral subject to certain boundary conditions. In 1933, Kantorovich proposed a method which reduced this problem to that of finding the minimum of a single integral. He chose part of the solution a priori (in accordance with the character of the problem) which left only one undetermined function in one of the variables. This unknown function could then be found through the solution to an ordinary differential equation. He then constructed a second approximation by introducing two unknown functions which led to the solution of two simultaneous differential equations. Again the accuracy of the solution was dependent on the initial choice for part of the function; so also was the rate of convergence.

Type
Technical Notes
Information
The Aeronautical Journal , Volume 71 , Issue 680 , August 1967 , pp. 573 - 575
Copyright
Copyright © Royal Aeronautical Society 1967

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Kantorovich, L. V. A Direct Method of Solving the Problem of the Minimum of a Double Integral. (Russian) 1 z V. AN SSSR OMEN. 1933.Google Scholar
2.Kantorovich, L. V. and Krylov, V. I.Approximate Methods of Higher Analysis. P. Noordhoflf Ltd, Gronissgen, The Netherlands, pp 304–325, 1958.Google Scholar
3.Sokolnikoff, I. S.Mathematical Theory of Elasticity, 2nd Ed, Chap 7, McGraw-Hill Book Co, New York. 1956.Google Scholar
4.Webber, J. P. H.Plane Stress Analyses of Rectangular Plates Subjected to Thermal Loading. Section 3, DPhil Thesis, University of Oxford, September 1966.Google Scholar
5.Poritsky, H.The Reduction of the Solution of Certain Partial Differential Equations to Ordinary Differential Equations. Proc 5th International Congress for Applied Mechanics. Cambridge, Massachusetts, 1938.Google Scholar